DEVELOPMENTS IN NONCOMMUTATIVE KOROVKIN-TYPE THEOREMS (Noncommutative Structure in Operator Theory and its Application)

DEVELOPMENTS

IN NONCOMMUTATIVE KOROVKIN-TYPE THEOREMS

M. N. N. NAMBOODIRI

Department ofMathematics,

Cochin University ofScience and Technology, Kochi-682022. Email: nambu@cusat.ac.in,nbdri@yahoo.com

1. INTRODUCTION

The classical approximation theorem dueto P. P. Korovkin [11] in 1953,

unified many existing approximation processes such

as

Bemstein

polyno-mial approximationofcontinuous functions. Korovkin’s discovery inspired many researchers that lead to Korovkin-type theorems and Korovkin sets in various settings such as

more

general function spaces, Banach algebras,

Banachlattices and operator algebras. Anothermajor advancementwas the discoveryofgeometric theory ofKorovkinsets by Y. A.

\v{S}a\v{s}kin

[21]and D.

E. Wulbert in 1968 [7]. A detailed survey of most of these developments

can

befound in the article of Berens and Lorentz in 1975 [7], monograph of Altomare and Campiti [1] _{most recent survey} by

Aitomare

[2] which

contains several

new

results also.

This article aims at providing arather short survey of the developments

in the so called noncommutative Korovkin-type approximation theoly and

Korovkin sets (quantization of Korovkin theorems, W. B. Arveson [6]) in

the settings of$C^{*}$ and $W^{*}$-algebras. Due to technical

reasons

only

impor-tant theorems are quoted that too without proofs. However

an

attempt is

made to provideillustrative examples, afew

new

results andresearch

prob-lems. First

we

quote three major theorems due to Korovkin following the

article ofBerens and Lorentz [7]. We designate these as Korovkin’s type

I, _{type II and type III theorems. The survey will be about ‘quantization’ of} these three theorems!

Type I Korovkin’s theorem. Let $\{\Phi_{n} : n=1,2,3, \ldots\}$ be a sequence of

positive linear maps on $C[a, b]$ and for each of the functions $g_{k}(x)=x^{k}$_, $x\in[a, b],$ $k=0,1,2$_, let

$\lim_{narrow\infty}\Phi_{n}(g_{k})=g_{k}$ uniformly on $[a, b],$ $k=0,1,2$

.

Then

$\lim_{narrow\infty}\Phi_{n}(f)=f$uniformlyon $[a, b]$ for all $f$ in $C[a, b]$.

Definition

1.1.

A set $S$ in _{$C[a, b]$} is called

a

test set

or

Korovkin set for

M. N. N. Namboodiri

linearoperators

on

$C[a, b],$ $\lim_{narrow\infty}\Phi_{n}(s)=s$uniformly

on

$[a, b]$ for

every

$s$ in $S$implies that $\lim_{narrow\infty}\Phi_{n}(f_{=}f$uniformly of $[a, b]$ for all $f\in C[a, b]$

.

Type I theorem says that $\{1, x, x^{2}\}$ is atest set.

Type II Korovkin’s theorem. There is not test set for $C[a, b]$ consisting

only oftwofunctions. Thus the cardinality ofatest set isatleast 3.

TypeIn Korovkin’s theorem. A triple $\{f_{0}, f_{1}, f_{2}\}$ is a test set of $C[a, b]$

exactly when itis a$\check{C}eby\check{s}ev$ system

on

$a,$$b$].

This article is divided into four sections. The next three sections

are

devoted to noncommutative Type I, _Type II and Type III theorems. The last section contains

some

aspects of weak Korovkin type theorems andits geometricformulation.

2. TYPE I THEOREMS

This section deals with type I Korovkin theorems in the settings of $C^{*}-$

algebras. It is assumed thatthe $C^{*}$-algebras considered here

are over

com-plex numbers andalways contain identityunless otherwise specified. Deflnition 2.1. Let $d$ and $\mathscr{B}$ be complex $C^{*}$-algebras with identities $1_{d}$

and $1_{\mathscr{D}}$ respectively and let $T$ : $darrow \mathscr{B}$ beapositive linear contraction (a

linear map $T$ that

preserves

positivity and such that $T(1_{d})\leq 1_{\mathscr{D}})$

.

For a

subset $H$ of,Ofthe Korovkin closure_{$K_{+}(H, T)$} is defined

as

$\{a\in d|\lim_{\alpha}\Phi_{\alpha}(a)=T(a)$ forevery net $\{\Phi_{\alpha}\}_{\alpha\in I}$

ofpositivelinearcontractions from to $\mathscr{B}$ such that

$\lim_{\alpha}\Phi_{\alpha}(h)=T(h)$ forall $h$ in $H$

}

Here

convergence

considered is the

norm convergence

unless otherwise stated explicitly. It

seems

that the first noncommutative Korovkintype the-orem was due to W. B. Arveson in 1970 for $*$-homomorphisms where .Of

is $C(X)$, the $C^{*}$-algebra of all complex continuous functions on a

com-pact, Hausdorff space $X$. We recall this, being the first of its kind. Recall

that for a subset $H$ of _$C(X)$, the Choquet boundary $\partial_{H}^{+}(X)$ is defined

as

the set of all points $x$ in $X$ such that the evaluation functionals $\epsilon_{x}|H$ has

the unique positive linear extension $\epsilon_{X}$ to $C(X)$. Also the support $K_{T}$ of

$T$ : $C(X)arrow \mathscr{B}$isdefined

as

setof all$x$in $X$ such that $f(x)=0$ whenever

$T(f)=0$

.

2.1. Theorem. Let$H$beasubsetof_$C(X)$ containing 1 andlet$T:C(X)arrow$

$\mathscr{B}$ be a _$*$ homomorphism such that _{$K_{T}\subseteq\partial_{H}^{+}(X)$}

.

Then

$K_{+}(H, T)=$ $C(X)$.

Theproofoftheabovetheoremusesthelatticetheoretic properties ofthe

selfadjoint part of$C(X)$

.

It is to be recalled that the selfadjoint _pall of a

$C^{*}$-algebra is a lattice in the natural order if and only of.Of is

out a relation between ‘noncommutative Choquet boundary’ and

‘hyper-rigid subspaces’ (Korovkin sets) _{for general} $C^{*}$ algebras. In whatfollows a

brief sketch of the developments from 1970 to 2010 is provided. Important theoremsthat appearedin the articles published during theabove period by

various authors, areregarding the following two questions.

(1)_{When does the Korovkin closure has an algebraic stmcture}

(2)Whenis the Korovkin closure the full $C^{*}$ algebra. Mainly fourtypes of

maps

areconsidered in this settings. Alinearmap $\Phi$ : $arrow \mathscr{B}$is called (a) Positiveif$\Phi(x^{*}x)$ is positive for all $x\in A$

(b) _Schwarz

map

if$\Phi(x^{*}x)\geq\phi(x)^{*}\Phi(X)$ for all $x\in$

(c) _{Completely positive}if$\Phi^{(n)}$ :

$\otimes M_{n}(\mathbb{C})arrow \mathscr{B}\otimes M_{n}(\mathbb{C})$ is a positive

for all positive integers$n$, where $M_{n}(\mathbb{C})$ is the setof all _{$n\cross n$} matrices

over$\mathbb{C}$ and $\Phi^{(n)}$ is themap on

$\otimes \mathscr{B}$defined by

$\Phi^{(n)}(a_{ij})=(\Phi(a_{ij}))$, where $(a_{ij})\in\otimes M_{n}(\mathbb{C})$

(d) _{Completely contractiveif}$\Phi^{(n)}$ is contractive foreachpositiveinteger

$n$

.

The main toolsbeing usedinthedevelopment of the commutativetheory

for positive linearmaps areKadison-SchwarztypeinequalitiesandChoquet boundary theory. Itistobementionedthateverycompletelypositivemapof

norm

$\leq 1$ is

a

Schwarz

map

and theverydefinitionofSchwarzmapimplies

the Schwarz type inequality. Forgeneral positive linearmap $\Phi$ : $arrow \mathscr{B}$

with

norm

$\leq 1$ Kadisonproved that

$\Phi(x^{2})\geq\Phi(x)^{2}\forall x\in,$$x^{*}=x$.

This fundamental inequality is known as Kadison-Schwarz inequality and

has been improved by many mathematicians like M. D. Choi [9] and T.

Fumta [10]. _These improvements will have some effect on the study of

Korovkinsets. Howeverthis possibility isyet to beinvestigated.

Using Kadison-Schwarz inequality, W. M. Priestley [17] provedthe

fol-lowingtheorem in 1976.

2.2. Theorem. Let bea$C^{*}$-algebraandlet $\{\Phi_{\alpha}\}_{\alpha\in I}$ beanetofpositive

linear

maps

on

such that

$\Phi_{\alpha}(1_{\ovalbox{\tt\small REJECT}})\leq 1_{\ovalbox{\tt\small REJECT}}$ $\forall\alpha\in I$.

Then the set

$J$ _{$:= \{x\in|\lim_{\alpha}\Phi_{\alpha}(a)=a\forall a\in\{x, x^{*}\circ x, x^{2}\}\}$}

isa $J^{*}$-algebra in .

Recall that a $J^{*}$-algebra in is a norm closed, $*$ closed subset of

whichis also closed under the Jordan product $0$, namely

$a\circ b=ab+ba,$ $a,$$b\in$

M. N. N.Namboodiri

2.3.

Theorem. Let be

a

$C^{*}$-algebra with identity $1_{d}$ and let $\{\Phi_{\alpha}\}_{\alpha\in I}$

be

a

net of Schwarz

maps

on

such that

$\Phi_{\alpha}(1_{d})\leq 1_{d}$, $\forall\alpha\in I$.

Then the subset

$K= \{x\in|\lim_{\alpha}\Phi_{\alpha}(a)=a$for$a\in\{x,$$x^{*}x,$ $xx^{*}\}\}$

is a $C^{*}$-algebra in$d$.

Subsequently B. V. Limaye and M. N. N. Namboodiri [12] _{improved the}

results ofPriestley andRobertson toobtain the following theorem in

1982:

2.4.

Theorem. Let and $\mathscr{B}$ be $C^{*}$-algebras with identities $1_{d}$ and $1_{\mathscr{D}}$

respectively. Let $\{\Phi_{\alpha}\}_{\alpha\in I}$ be

a

net of positive linear maps from $d$ to $\mathscr{B}$

such that

$\Phi_{\alpha}(1_{d})\leq 1_{\ovalbox{\tt\small REJECT}}$ $\forall\alpha\in I$.

Let $T$ : $arrow \mathscr{B}$be_$a*$-homomorphism. Then the subset

$J$ _{$:= \{x\in|\lim_{\alpha}\Phi_{\alpha}(a)=T(a)$}, for$a\in\{x,$$x^{*}ox\}\}$

is a $J^{*}$-algebra in.Of.

If all $\Phi_{\alpha},$ $\alpha\in I$

are

Schwarz maps and $Ta*$-homomorphism, then $J$ is

a $C^{*}$-algebra in .

In the above

cases

thetest set

was

symmetricwith respectto $*$ operation.

For the

case

when this symmetry is not assumed, B. V. Limaye, M. N. N. Namboodiri in 1984 [14] and A. G. Robertson in 1986 [19] _proved the followingtheorems.

It is known thatKorovkin type approximation theory leads to deeper

un-derstanding ofthe stmcture under consideration. The following interesting

theoremin [14] reveals this.

2.5. Theorem. Let$=\mathscr{B}$ be

a

noncommutative $C^{*}$ algebra withidentity

$1_{d}$ and let

$D= \{x\in d|\lim_{\alpha}\Phi_{\alpha}(a)=a$for all $a\in\{x,$ $x^{*}x\}\}$

where $\{\Phi_{\alpha}\}_{\alpha\in I}$ is anet ofSchwarz maps on$d$ with norm $\leq 1$. Then $D$is

a subalgebra of$d$

.

Also$D$ is $*$ closed if and only if .Of $=M_{2}(\mathbb{C})$, the set

ofall $2\cross 2$ matrices

over

$\mathbb{C}$.

The above theorem shows that $M_{2}(\mathbb{C})$ behaves like a commutative $C^{*}-$

algebra and this is the only noncommutative one! In fact, Limaye and

Namboodiri proved that,

among

allfinite dimensional noncommutative $C^{*}$

algebra $M_{2}(\mathbb{C})$ is the only one for which $D$ in Theorem 2.5 is $*$ closed.

Robertsonproved that finitedimensionalityassumption

can

bedropped.

We bypass several important developments ofKorovkin approximation

theory for commutative as well

as

noncommutative Banach algebras.

Ex-cellent exposition

can

be found by Micheal Panneberg, [1, _Appendix $A$],

Ferdinand Beckhoff [1, Appendix $B$] and F. Altomare [2]. However M.

al

so

obtains several extensionsof Korovkintypetheorems byusingoperator

monotone functions and T. Ando’s inequality. In this paperhe also unified

severalearlierresults byintroducing $0^{*}$-subalgebras and the associated

gen-eralized Schwarz maps with respect to theproduct$0$. More overtheproofs

that he gives are simplerthan the earlierones.

For the sake of completion

we

quote acouple oftheorems of Uchiyama

[23] _forthe $C^{*}$ algebra $C(X)$.

2.6. Theorem. [23, _Theorem 3.1]

Let $S\subset C(X)$ and $C^{*}(S)$ be the $C^{*}$-algebra generated by $S$. Let $f$ be

a

operator monotone function defined

on

$[0, \infty]$ such that _{$f(O)\leq 0$} and

$f(\infty)=\infty$

.

Set $g=f^{-1}$ then

we

have

$C^{*}(S)\subseteq K_{C(X)}(SU\{g(|u|^{2}) : u\in S\})$

if$f(0)=0$ or $1\in S$. Where the set on the rightside denotes theKorovkin

closure.

2.7. Theorem. [23, _Theorem 2.12] _Let $\{\Phi_{n}\}$ be a sequence of Schwarz

maps from to $\mathscr{B}$ where and $\mathscr{B}$ are $C^{*}$-algebras with identities, and

let $\Phi$ : _{$arrow \mathscr{B}$} be

$a*$-homomorphism. Let $f$ be an operator monotone

function on $[0, \infty]$ with $f(0)=0,$ $f(\infty)=\infty$

.

Set $g=f^{-1}$. Then the set

$C=\{a\in|\Phi_{n}(x)arrow\Phi(x)$ for _$x=a,$$g(a^{*}a)$ and $g(aa^{*})\}$

is a $C^{*}$-subalgebra.

Another important development

was

the

use

of Krein-Millman theorem

forcompact

convex

sets and the associatedunique extension property. For

noncommutative $C^{*}$-algebras, the following theorem was proved by Taka-hasiin 1979 [22].

2.8. Theorem. Let$\phi$be

an

extremestateofa$C^{*}$ algebra andlet_{$x\in_{+}$}

peaksfor$\phi$, that is the supports of_$x$ and$\phi$in the enveloping

von

Neumann

algebra add up to 1. Let $\{\phi_{\alpha}\}_{\alpha\in I}$ bea net ofpositive linearfunctionals of

and

assume

that $\lim_{\alpha}\phi_{\alpha}(1-)=1$ and $\lim_{\alpha}\phi_{\alpha}(x)=0$

.

Then we

have

$\lim_{\alpha}\phi_{\alpha}(a)=0$forall $a$om .Of.

Though the above theoremiselegant, for

an

arbitrary$C^{*}$-algebras thereis

no

way offinding extreme states (or

pure

states), _where

as

_{forcommutative}

$C^{*}$ algebras extreme states are point evaluations. In the caseof $C^{*}$-algebra

$B(H),$ $H$ aHilbert space, $\phi$a vector statebetter results

are

known to exist.

See for example Limaye-Namboodiri (1979) [1, Appendix B.], _Altomare

1987 [1] and Dieckmann 1992 [1].

2.9. Theorem (Altomare [1]). Let $T\in B(H)$ , be a non

zero

compact

operator, $\lambda$asimpleeigenvalue of_$T$such that

$\Vert T\Vert=|\lambda|,$ $x$

a

corresponding

unit eigenvectorand let $\phi=\langle\cdot x,$$x\rangle$ be thevectorstate correspondingto _$x$.

Put $S=I_{H}+ \frac{1}{|\lambda|^{2}}T^{*}T-\frac{1}{\lambda}T_{\lambda}^{1}-arrow T^{*}$. Then

M. N. N.Namboodiri

Here $I_{H}$ denotes the identity operator

on

$H$

.

The following theorem is due to Dieckmann

2.10. Theorem (Dieckmann). Let $T\in B(H)$ be a strictly positive

com-pact operator

on

$H$ and let $d(H)=C^{*}(I_{H}, K(H))$, where $K(H)$ is the

setofall compact operators

on

$H$. Let$\phi$bethecomplex homomorphism

on

$(H)$ defined by$\phi(\lambda I_{H}+K)=\lambda,$ $K\in K(H)$. Then $K_{+}(\{I_{H}, T\}, \phi)=$

$(H)$

.

Finally we go through Arveson’s contributions to noncommutative

Ko-rovkin type theorems and Korovkin sets in 2009 [6] via his

own

theory of

noncommutativeChoquet boundary theory of operator systems. In the fun-damental

papers

during

1969-70

and2008, he introduced and proved

many

concepts and the theorems related to

non

commutative Choquet boundary and Silov boundaryideals $co$lTespondingto operator systems. This isquite

analogous to classical theory of Choquet and Silov boundaries for

func-tion systems. Analogous to the work ofSaskin, Arveson studied the

rela-tionbetween noncommutative Korovkin setsandnoncommutativeChoquet

boundaryin2009 [6]. Heprovedmany interestingtheoremsinthis settings,

though

some

of theseresults

were

already knownto exist. Westartwith the

notion ofhyperrigid setofgenerators of$C^{*}$ algebras [6].

2.11. Definition. A finite ofcountablyinfiniteset$G$of generatorsof$C^{*}$

al-gebra.Of is said to behyperrigidiffor

every

faithfulrepresentation $\pi()\subseteq$

$B(H)$ of al

on

a

Hilbert

space

$H$ and _$evel\gamma$

sequence

of unit preserving

completelypositivemaps (UCP) $\Phi_{n}$ : $B(H)arrow B(H),$ $n=1,2,3,$

$\ldots$

$\lim_{narrow\infty}\Vert\Phi_{n}(\pi(g)-\pi(g))\Vert=0\forall g\in G\Rightarrow\lim_{narrow\infty}\Vert\Vert\Phi_{n}(\pi(a)-\pi(a))\Vert=0$,

$\forall a\in$

.

He then proves the followingbasic theorem.

2.12. Theorem. For $evel\gamma$ separable operator system $S$ that generates

a

$C^{*}$-algebra , the following

are

equivalent.

(i) $S$is hypemigid

(ii)For$evel\gamma$

non

degeneraterepresentation $\pi$ : $arrow B(H)$

on

aseparable

Hilbert space$H$and every

sequence

$\Phi_{n}$ : $arrow B(H)$

of UCPmaps;

$\lim_{narrow\infty}\Vert\Phi_{n}(s)-\pi(s)\Vert=0$ $\forall s\in S\Rightarrow\lim_{narrow\infty}\Vert\Phi_{n}(a)-\pi(a)\Vert=0$

for all $a\in d$

.

(iii)Foreverynondegeneraterepresentation$\pi$ : $arrow B(H)$ on aseparable

Hilbert space, $\pi/S$ has the unique extension property. That is, $\pi/S$ has a

uniquecompletely positive linear extensionto ,Of.

(iv) For every unital $C^{*}$ algebra $\mathscr{B}$, every unital homomorphism of $C^{*}-$

algebras $\theta$ : _{$arrow \mathscr{B}$}andeveryUCP map $\Phi$‘ : $\mathscr{B}arrow \mathscr{B}$

$\Phi(x)=x$ $\forall x\in\theta(S)\Rightarrow\Phi(x)=x\forall x\in\theta(d)$.

One ofthemain results (Theorem 3.3 [6]) that Arveson obtainsas a

can

befound in [6]. _{However he}

proves

_a $vel\gamma$ strongtheorem [6, Theorem

5.1] _whichis

as

follows.

2.13. Theorem. Let $S$ be a separable operator system whose generated

$C^{*}$-algebra,Ofhas countablespectrum such that

$evei\gamma$irreducible

represen-tation of is aboundary representationfor$S$. Then $S$ is hyperrigid.

2.14. Arveson’s conjecture. Ifevery irreducible representation of is a boundaryrepresentation foraseparable operator system $S$,then$S$ is

hyper-rigid.

Now recall that the following theorem was proved by Y. A. Saskin for

positive linear contractions and D. E. Wulbert for linear contractions [7].

2.15. Theorem. Let $G$ be a subset of _$C(X)$ that separates points of $X$

andcontains the constantfunction $1_{X}$. Then $G$is a Korovkin set for linear

contractions orpositivelinearcontractions if and only if$\partial_{Ch}G_{0}=X,$_{$G_{0}=$}

span$G$.

The noncommutative Choquet boundary

was

defined by Arveson [6] in

thefollowingway.

2.16. Definition. Let $S$ be

an

operator system in a $C^{*}$-algebra , i.e., a

self adjoint linear subspace of such that $1_{d}\in S$ and $=C^{*}(S)$

the $C^{*}$-algebra generated by $S$ and $1_{d}$. A boundary representation for $S$

is

an

$i_{lT}educible$ representation $\pi$ of such that $\pi/S$ has a unique

com-pletely positive linear extensionto $d$. The set$\partial_{S}$ of all unitary equivalence

classes of all boundary representations for $S$ id defined

as

the

noncommu-tative Choquet boundary ofthe operator system $S$.

Since$i_{lT}educible$representation of thefunction space _$C(X)$ canbe

iden-tified with points in $X$ itself, Arveson’s notion of Choquet boundary for

operatorsystems is

an

exact noncommutativeanalogue of the classical one

forfunction systems.

In what follows we examinethe possibility of extending Arveson’s

theo-rem

quoted here for linearcontractions. Since extension theorem for

com-pletely positive

maps

is not available,

we

needto define hypernigidity

sep-arately. So

we

introduce strong hyperrigidity

so as

to suitcompletely con-tractive

maps.

2.17. Definition. A finite

or

countably infinite set $G$ofgenerators ofa$C^{*}-$

algebra is said to be strongly hyperrigid if for every faithful

represen-tation $\pi$ of in $B(H)$ and for every sequence $\Phi_{n}$ completely contractive

maps

from $\pi()$ to $B(H)$

$\lim_{narrow\infty}\Vert\Phi_{n}(\pi(g))-\pi(g)\Vert=0$ $\forall g\in G$

M. N.N.$Nambood\ddot{m}$

2.18. Remarks. It

can

be

seen

that the strong hyperrigidity coincide with

hyperrigidity for UCP

as a consequence

of Arveson’sextensiontheorem for CP

maps.

Howeverhyperrigidity of$G$neednot imply strong hyperrigidity.

To

overcome

this difficultyitwould be reasonableto

assume

that$G$is closed

under $*$ operation ifnecessary.

In what follows

we

aimatidentifying ‘obstmctions’ to strong hyperrigid-ity. We also

assume

that theoperator system is $*$ closed and contains

iden-tity element.

2.19. Characterisation theorem. For separable operator system $S$ that

generates a $C^{*}$-algebra ,Of, thefollowingare equivalent:

(i) $S$ is strongly hyperrigid.

(ii) For

every

non

degenerate representation $\pi$ : $arrow B(H)$

on

a

sep-arable Hilbert

space

$H$ and

every sequence

$\Phi_{n}$ : $darrow B(H)$ completely

contractive maps,

$\lim_{narrow\infty}\Vert\Phi_{n}(s)-\pi(s)\Vert=0$ $\forall s\in S$

$\Rightarrow\lim_{narrow\infty}\Vert\Phi_{n}(a)-\pi(a)\Vert=0$ $\forall a\in$.

(iii) Forevery

non

degeneraterepresentation $\pi$ : $darrow B(H)$ on a

sepa-rable Hilbert space $H,$ $\pi/S$ has theuniqueextension property.

(iv) For$evei\gamma$ unital $C^{*}$-algebra $\mathscr{B}$, every unital homomorphism of$C^{*}-$

algebras $\theta$ : $darrow \mathscr{B}$ andevery (forevery UCPmap $\Phi$ : _{$\theta(d)arrow \mathscr{B}$}) map

$\Phi\thetaarrow \mathscr{B}$

$\Phi(x)=x$ $\forall x\in\theta(S)\Rightarrow\Phi(x)=x$ $\forall x\in\theta(d)$.

Proof.

Theproof is more orless the

same

as thatofArveson. However the

details

are

provided. We show that

$(i)\Rightarrow(ii)\Rightarrow(iii)\Rightarrow(iv)\Rightarrow(i)$

.

$(i)\Rightarrow(ii)$

Let $\pi$ : $arrow B(H)$ be a

non

degenerate representation of

on

a

separable Hilbert space $H$ and let $\Phi_{n}$ : $darrow B(H)$ be a sequence of

(completely contractive) linearmaps such that

$\lim_{narrow\infty}\Vert\Phi_{n}(s)-\pi(s)\Vert=0$ $\forall s\in S$.

Let $\sigma$ : $arrow B(H)$ beafaithful representation of

on

another separable

Hilbert space $K$.

Then $\sigma\oplus\pi$ : $arrow B(K\oplus H)$ is a faithful representation of $d$

on

$K\oplus H$. Define maps $\mu_{n}$ : $(\sigma\oplus\pi)(d)arrow B(K\oplus H)$ by $\mu_{n}(\sigma(a)\oplus\pi(a))=\sigma(a)\oplus\Phi_{n}(a)$, $a\in$.

Then $\mu_{n}$ is completelycontractive.

Also $\mu_{n}(\sigma(s)\oplus\pi(s)arrow\sigma(s)\oplus\pi(s))$, for all $s\in S$.

Now,

$\lim_{narrow\infty}\sup\Vert\Phi_{n}(a)-\pi(a)\Vert\leq\lim_{narrow\infty}sub\Vert\sigma(a)\oplus\Phi(a)-\sigma(a)\oplus\pi(a)\Vert$

$=$ sub$\Vert\mu_{n}(\sigma(a)\oplus\pi(a))-\sigma(a)\oplus\pi(a)\Vert$

Therefore

$\lim_{narrow\infty}\Vert\Phi_{n}(a)-\pi(a)\Vert=0$ $a\in$.

Now $(iii)\Rightarrow(iv)$: Let$\theta$ : _{$arrow \mathscr{B}$} be

an

identity preserving homomorphism

of$C^{*}$-algebras andlet $\Phi$ : $arrow \mathscr{B}$ be

a

UCP that satisfies _{$\Phi(\theta(s))=\theta(s)$},

$s\in S$

.

We have to show that

$\Phi(\theta(a))=\theta(a)$ $\forall a\in$ 2.

Let $B_{0}$ bethe separable $C^{*}$-algebrain $\mathscr{B}$ generated by

$\theta()\cup\Phi(\theta())\cup\Phi^{2}(\theta())\cup\cdots$

It is clearthat $\Phi(B_{0})\subseteq B_{0}$

.

By considering afaithful representation of$B_{0}$ on a separableHilbertspace

$H$,

we

may

assume

that _{$B_{0}\subseteq B(H)$}

.

Let$\tilde{\Phi}$

: $B(H)arrow B(H)$isaUCPmap

either$\overline{\Phi}/B_{0}=\Phi$

.

Here $\overline{\Phi}(\theta(s))=\theta(s),$ $\forall s\in S.$ $S$ince $\theta$ : $arrow B(H)$ is

arepresentationon $H$, we must have

$\Phi(\theta(a))=\overline{\Phi}(\theta(a))=\theta(a)$ $\forall a\in A$.

Hencethe proof.

$(iv)\Rightarrow(i)$ Let $\pi$ : $arrow B(H)$ be afaithful representation of on $B(H)$

for

some

Hilbert space $H$. Put _{$\mathscr{B}=B(H)$}. Consider the $C^{*}$-algebras of

all bounded seqences $l^{\infty}()$ and $l^{\infty}(\mathscr{B})$ in and $\mathscr{B}$ respectively. Let

$\Phi_{n}$ : $\pi(A)arrow B(H)$ be the

sequence

of all completely contractive maps

such that

$\lim_{narrow\infty}\Vert\Phi_{n}(\pi(s)-\pi(s))\Vert=0$ $\forall s\in S$.

To show that

$\lim_{narrow\infty}\Vert\Phi_{n}(\pi(a)-\pi(a)\Vert>0$ $\forall a\in$.

Let

$\tilde{\Phi}$ :

$l^{\infty}()arrow l^{\infty}(\mathscr{B})$be defined

as

$\tilde{\Phi}(a_{1}, a_{2}\ldots a_{n}, \ldots)=(\Phi_{1}(a_{1}), \Phi_{2}(a_{2}), \ldots)$

$(a_{1}, a_{2}, \ldots, a_{n}, \ldots)\in l^{\alpha}()$

First we show that $\tilde{\Phi}$

is completely contractive. Itis quite easy to

see

that

$l^{\infty}()\otimes M_{n}(C)$

can

be identified isometricallywith$l^{\alpha}(\otimes M_{n}(C))$. The

same

way identify $l^{\infty}(\mathscr{B}\otimes M_{n}(C))$ with $l^{\infty}(\mathscr{B})\otimes M_{n}(C)$.

Thus $\tilde{\Phi}^{(n)}$

can

be regarded as a

map

from $l^{\infty}(\otimes M_{n}(C))$ to $l^{\infty}(\mathscr{B}\otimes$

$M_{n}(C))$, foreach positive integer$n.\tilde{\Phi}^{(n)}$ is themap inducedby $\tilde{\Phi}$

for each

M. N.N.Namboodiri

It is easy to

see

that $\tilde{\Phi}^{(n)}$

is contractive since $\Phi^{(n)}$ is contractive and

$\Phi^{(n)}(\pi(I))arrow\pi(I)$

as

$narrow\infty$.

Let $C_{0}()$ (respectively $C_{0}(\mathscr{B})$) denotes the ideal of all

sequences

in $\mathscr{A}$

(respectively $\mathscr{B}$) that

converges

to

zero

in

norm.

Consider themap

$\tilde{\Phi}_{0}:\frac{l^{\infty}()}{C_{0}()}arrow\frac{l^{\infty}(\mathscr{B})}{C_{0}(\mathscr{B})}$

defined by

$\tilde{\Phi}_{0}(x+C_{0}())=\tilde{\Phi}(x)+C_{0}(\mathscr{B})$, $x\in l^{\infty}()$

Then $\tilde{\Phi}_{0}$ is completely contractive. Consider the embedding $\theta$ : $arrow$

$l^{\infty}(d)$ defined by

$\theta(a)=(a, a, \ldots)+C_{0}(d)$

Therefore$\tilde{\Phi}_{0}(\theta(s))=(\Phi_{1}(s), \Phi_{2}(s), \ldots)+C_{0}()$

$=(s, s, \ldots, s, \ldots)+C_{0}(d)$

$=\theta(s)$ $\forall s\in S$

Thus

$\tilde{\Phi}_{0}:\frac{l^{\infty}(d)}{C_{0}()}arrow\frac{l^{\infty}(\mathscr{B})}{C_{0}(\mathscr{B})}$

such that

$\tilde{\Phi}_{0}(\theta(s))=\theta(s)$ $s\in S$. $\Rightarrow\tilde{\Phi}_{0}$

is

a

UCP since identity $1_{d}\in S$

.

This is because, if and $\mathscr{B}$

are

$C^{*}-$

algebras with identities $1_{d}$ and $1_{\ovalbox{\tt\small REJECT}}$ and if $\Phi$ : $darrow \mathscr{B}$ is a contractive

linear

map

such that

$\Vert\Phi(1_{d})\Vert=\Vert\Phi\Vert$,

then $\Phi$ ispositivity preserving. Then $\tilde{\Phi}_{0}(\theta(a))=\theta(a)$ for all $a\in$

.

That

is,

$(\Phi_{1}(a), \Phi_{2}(a), \ldots)+C_{0}(d)$

$=(a, a, \ldots)+C_{0}()$ $a\in$

$\Rightarrow\Vert\Phi_{n}(a)-a\Vertarrow 0$ as $narrow\infty$

$\square$

2.20. Remarks. The abovetheoremis

a

noncommutativeanalogueof

Wul-berts theorem for hyperrigidity in function spaces such as $C(X)$

.

This is

because every contractive linear map on $C(X)$ is completely contractive

[3, 4]. So most of Arveson’s theorem for hyperrigidity for $C^{*}$ algebras is

valid for strong hyperrigidityalso. We state

some

ofthese without proof. 2.21. Corollary. Let $S$beastronglyhyperrigidseparable operatorsystem,

with generated$C^{*}$-algebra $d_{1}$ let$K$ be

an

ideal in and let$a\in\mapsto\dot{a}\in$

$d/K$be thequotientmap. Then $\dot{S}$is astronglyhyperrigid operator system

2.22. Theorem. Let $x\in B(H)$ be

a

self adjoint operator with atleast 3

points in its spectmm and let be the $C^{*}$-algebra generated by _$x$ and 1.

Then

(i) $G=\{1, x, x^{2}\}$ is astrongly hypeiTigid operatorsystemfor _, while (ii) $G_{0}=\{1, x\}$ is nota strongly hyperrigidgenerator for.Of.

2.23. Theorem. Let $\{u_{1}, u_{2}, \ldots u_{n}\}$ be a setofisometricesthat generatea

$C^{*}$-algebra,Of and let

$G=\{u_{1}, u_{2}, \ldots u_{n}, u_{1}^{*}, u_{1}+u_{2}^{*}u_{2}+\cdots+u_{n}^{*}u_{n}\}$ .

Then $G$is astrongly hyperrigid generatorfor .

2.24. Corollary. Theset $G=\{u_{1}, u_{2}, \ldots, u_{n}\}$, where$\sum_{k=1}^{n}u_{k}u_{k}^{*}=I$, of

generators of the Cuntz algebra$\theta_{n}$ is strongly hyperrigid.

We conclude this section by remarking that many more implication of

‘

strong hyperrigiditytheorem’ are tobe investigated. However such results

will appearelsewhere.

3. TYPE II KOROVKIN THEOREMS

Recall that this section deals with thesize ofatest set $H$in a $C^{*}$-algebra

.Of generated by $H$. In $C[a, b]$_, there is no _{test set} containing only _two

elements.

Observe that (i) _{of2.22 is} already known, whereas (ii)does not seemto

existin this generality. Does this result have a noncommutativeanalogue?

3. 1. Question. Let$x\in B(H)$. Let bethe$C^{*}$-algebra generated by $I$ad

X. Then itisknown that $\{I, x, x^{*}x+xx^{*}\}$ is hyperrigid in . If spectmm

$\sigma(x)$ has atleast3 distinct points,thenisittme that $\{I, x\}$isnot

a

hyperrigid

generatorof ?

The following simplemodification of2.22is possible.

3.2. Proposition. Let $x\in B(H)$ be normal. Then $G=\{1, x, x^{*}x\}$ is a

hyperrigid set of generators for ,Of $=C^{*}(x)$. If $\sigma(x)$ contains three

dis-tinct points $\lambda_{1},$ $\lambda_{2}$ and $\lambda_{3}$, on some straight line, then $\{$1,_$x\}$ will not be a

hyperrigidgeneratorfor .

We provide the proof for the sake of completion.

Proof.

Statement (i) _is already known. Now we

prove

(ii) using Arveson’s argument. Let $S=$ span$\{$1,_$x\}$ and let$\sigma(x)$ denote the spectmm of$x$. For

$f\in C(\sigma(x))$, let $\phi_{k}(f(x))=f(\lambda_{k}),$ $k=1,2,3$.

Then $\phi_{k}$ is amultiplicative positive linear functional of

norm

1 which is

an

irreducible representation of on $\mathbb{C}$. But

$\phi_{k}(\lambda 1_{\alpha}+\mu_{x})=\lambda+\mu\phi_{k}(x)$

$=\lambda+\mu\lambda_{k},$ $k=1,2,3$.

But $\lambda_{2}$ is a convex combination (assume without loss of generality) of $\lambda_{1}$

M. N. N.Namboodiri

Therefore $\phi_{2}(\lambda 1_{d}+\mu x)=t\phi_{1}(\lambda 1_{d}+\mu x)+(1-t)\phi_{3}\lambda 1_{d}+\mu x)$

.

Thus the positive linear functional $\phi=t\phi_{1}+(1-\phi)\phi_{3}$ and $\phi_{2}$

are

two

different completely positiveextensions of$\phi_{2}/S$. Therefore the irreducible

representation $\phi_{2}$ fails to haveunique extension property therefore

one

$x$is

not hyperrigid. $\square$

Weconcludethis sectionby statinga problem of Arveson [6].

3.3. Question. Let $I=[a, b],$ $f$ : $Iarrow R$and$A\in B(H)$ be selfadjoint. Is

$[1, A, f(A)]$ hyperrigid in $C^{*}(A)$? Arveson observesthat in case $A$has

dis-crete spectmm in $[a, b]$ and if$f$is either strictly convex or strictly concave,

the

answer

is affirmative.

4. TYPE III KOROVKIN THEOREMS

Recall theclassical Korovkin theorem says that $\{f_{1}, f_{2}, f_{3}\}$ is hyperrigid

in $C[a, b]$ exactly when span $\{f_{1}, f_{2}, f_{3}\}$ is a $\check{C}eby\check{s}ev$ system. It would

be interesting to examine its

non

commutative counterpart using $\check{C}eby\check{s}$ev $\vee$

systems in $C^{*}$-algebra. First we recall the notion of Ceby\v{s}ev system in

Banach spaces.

4.1. Definition. Let $M$ be a subspace of a Banach space. $N$ is called a

$\check{C}eby\check{s}ev$ systemifeach vector $N$admits aunique closestpoint in $M$

.

A. Haar in 1918 [20] _{obtained the following characterization of} finite dimensional $\check{C}eby\check{s}ev$ subspaces of $C(X),$ $X$ compact and Hausdorff. For

$C^{*}$ algebras the study

was

camiedout by A. G. Robertson, David Yost and

G. K. Pederson [16]

4.2. Proposition. [7]Let$M$beann-dimensionalsubspaceof$C(X)$. Then

$M$is a $\check{C}eby\check{s}ev$ system if and only ifno non

zero

function in _$M$ has

more

that$n-1$

zeros.

4.3. Theorem. [7] Let$X$denote

an

interval $[a, b]$

or

the unit circle$T$. Then

each $\check{C}eby\check{s}ev$ system _{$S=\{g_{0}, g_{1}, \ldots, g_{m}\}m\geq 2$} is a Korovkin set

(hy-pernigid)

It is to be remarkedthat finite Korovkinsets have been studied for

func-tion spaces, commutative Banach algebras and for

some

special types of

$C^{*}$ algebras [1]. We pose the following problem whose

answer

is not yet

known.

4.4. Question. Let $M$ be

an

$n$ dimensional subspace of $C^{*}$ algebras ,

where $n\geq 3$. Isit tmethat $M$is hyperrigid if itis a$\check{C}eby\check{s}ev$ subspace?

We conclude this section by mentioning few things regarding weak

Ko-rovkin type theorems.

When approximation in the weak

sense

by completely positive linear maps on $B(H)$ is considered, Korovkin type results have been obtained

in [13]. _Forexample, recall the definition ofweak Korovkin set introduced in [13]. It is asfollows:

4.5. Definition. A subset $S$ of $B(H)$ is called a weak Korovkin set if for

each net $\Phi_{\alpha}$ ofcompletelypositive

maps

satisfying _{$\Phi_{\alpha}(I)\leq I$}, therelation

$\Phi_{\alpha}(s)arrow s$ weakly, $s\in S$implies $\Phi_{\alpha}(T)arrow T$ weakly _{$T\in B(H)$}.

One of themaintheorems proved in [13] is

as

follows.

4.6. Theorem. Let $S$ be

an

irreducible set $B(H)$ such that $S$ contains the

identity operator$I$ and_$C^{*}(S)$ containsa

non

zerocompactoperator. Then $S$

is a weakKorovkin set in $B(H)$ if and only if id$|s$ has a unique completely

positive linear extension to $C^{*}(S)$ namelyid$|_{C^{*}(S)}$.

4.7. Remarks. The condition ‘id$|s$ has a unique completely positive

lin-ear

extension to $C^{*}(S)$’ means that the identity representation of $C^{*}(S)$ is

boundalyrepresentation for$S$ in the

sense

of Arveson.

Thefollowingboundary theorem ofArvesonenablestoidentifyanumber

of weak Korovkin sets.

4.8. Boundary theorem ofArveson. Let $S$ be a irreducible set in $B(H)$

such that $S$ contains the identity operator and $C^{*}(S)$ contains a

non

zero

compact operator. Then the identity representation of $C^{*}(S)$ is a bound-ary representation for $S$, if and only if the quotient map $q$ : $B(H)arrow$

$B(H)/K(H)$ is not completely isometric

on

span$(S+S^{*})$ where $K(H)$

denote the setof all compact operators

on

$H$

.

One oftheimplications provides thefollowing example.

4.9. Example. Let$S$beanirreducible operator whichisalmost normal but

notnormal, _{then the} set $S=\{I, s, s^{*}s+ss^{*}\}$ is a weak Korovkin set.

Acknowledgement. The author is thankful to the organisers of the RIMS

Kyoto Symposium, Oct.27-29, 2010 for the invitation

as

well

as

for local

hospitality;CUSAT andNBHM-DAE, Govt. of Indiaforfinancial support.

Also the authoris thankfulto Prof. B. V.Limaye forsuggesting

some

mod-ifications and corrections.

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